Physica A 273 (1999) 241–247
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Structural properties of a two-dimensional model for partly quenched colloidal dispersions Wojciech Rzyskoa;b , Orest Pizioa; ∗ , Stefan Sokolowskib a Instituto
b Department
de Qumica de la UNAM, Coyoacan 04510, Mexico, D.F., Mexico of Modelling of Physico-Chemical Processes, Marie Curie-Sklodowska University, Lublin 200-31, Poland Received 15 January 1999
Abstract We have investigated a two-dimensional model for a three-component mixture with a large di erence in diameters of hard core particles mimicking two colloidal species and in diameter of solvent species. One of the colloidal species is assumed to be quenched in disordered con guration and forms an adsorbing medium. Another colloidal species together with the solvent component attains an equilibrium distribution in a microporous space. The system is studied by using replica Ornstein–Zernike integral equations. Our main focus is on the behavior of the potential of mean force between movable colloids dependent on matrix and solvent density. c 1999 Elsevier Science B.V. All rights reserved.
Keywords: Structural force; Colloidal dispersion; Disordered medium
There have been extensive studies of colloidal dispersions in the framework of the model of hard spheres with di erent diameters, see e.g. Refs. [1–5]. It has been shown that the mean force potential (MFP) between large spheres exhibits depletion at small separations. Besides, the oscillations of the colloid–colloid MFP begin to develop with increasing solvent density. Some recent experimental e orts have been focused on the behavior of colloidal dispersions under con nement [6 –8]. In particular, the structural and dynamical properties of a two-component colloidal dispersion con ned within two glass plates, approaching each other, have been studied. One of the colloidal species becomes unmovable, if the separation of plates coincides with the larger colloid diameter. One of the most important ndings of the above research was an unusual shape of the MFP in dispersions ∗
Corresponding author. Fax: +52-5-616-2217. E-mail address:
[email protected] (O. Pizio)
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 2 9 8 - 8
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under con nement, as it follows from an analysis of experimentally observed (in the framework of the digital videomicroscopy) pair distribution functions [8]. The experimentally studied system is quite complicated for theoretical analysis. First, it is not exactly a two-dimensional (2D) system, rather it is a quasi-2D system. Second, the system in question is a colloidal dispersion consisting of charged particles con ned between charged plates. The solvent e ect on the observed pdf ’s also is dicult to elucidate from the available experimental data [8]. Due to these factors, at least, it is very dicult to apply an adequate model and obtain theoretically the MFP acting between colloids. Nevertheless, in order to get insight into the trends of behavior of the MFP between annealed colloids, in Ref. [8], the Ornstein–Zernike (OZ) 2D integral equations for a mixture of two colloidal species has been used. It is worth mentioning, that the closest, partly quenched inhomogeneous model and relevant theoretical approach for the system of experimental focus, has been developed and applied recently in Refs. [9 –11]. However, even under a simplifying two-dimensional approximation for actual quasi-2D system, the methodologically correct integral equations for partly quenched systems are the replica OZ (ROZ) equations developed by Given and Stell [12–14] on the basis of previous, pioneering studies of Madden and Glandt in this area [15,16], but not the usual OZ used in Ref. [8]. In the present communication we would like to initiate a systematic study of the adsorption and structural properties of colloidal systems in disordered matrices. To begin with, we are considering a two-dimensional model consisting of two components of hard discs mimicking colloidal particles and hard discs mimicking solvent species. One of the colloidal species of larger diameter is considered to be quenched in an equlibrium con guration. Our main focus is the structural properties of the system in terms of the pair distribution functions and the MFP for movable (annealed) colloids. With this aim we apply Ornstein–Zernike equations complemented by the Percus– Yevick closure approximation. Let us consider a two-component (c and s) uid of discs adsorbed in a disordered quenched matrix (m) of discs. We choose the diameter of solvent discs as a length unit, s = 1. The diameter of matrix and colloid species is denoted by m and c , respectively. The model for interparticle interactions is the following: ∞; R ¡ ij ; Uij (R) = (1) 0; R ¿ ij ; where ij = (i + j )=2 and i; j take values c; s; m. The microporous medium is generated by considering a matrix as an equilibrium con guration of hard discs, each of diameter m ; at density m . The matrix microporosity, p; is de ned as a fraction of volume accessible to accommodate uid species, p = 1 − m ; m = m m2 =4. Let us de ne the Mayer functions, fij (R); fij (R) = exp[ − Uij (R)] − 1 ; necessary for the application of integral equations theory.
(2)
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In the case of a matrix made of hard discs we describe the matrix structure, in terms of the pair correlation function of matrix species hmm ; using common OZ integral equation for a single-component uid of discs, hmm − cmm = m cmm ⊗ hmm
(3)
with the Percus–Yevick closure cmm (R) = {exp[ − Umm (R)] − 1}{1 + hmm (R) − cmm (R)} :
(4)
In Eq. (3) we have omitted R-dependencies for the sake of brevity. Also, the symbol ⊗ denotes convolution in two-dimensional space. Similar notations and abbreviation will be used below. The ROZ equations for the model in question have the form X him − cim = l cil ⊗ hlm ; l=c; s
hij − cij = m cim ⊗ hjm +
X
l cil ⊗ hlj ;
(5)
l=c; s
where i; j take values s and c. The above equations have been written in the form consistent with Madden–Glandt (MG) approximation [15,16], i.e. the blocking part of the direct correlation function for uid species has been assumed vanishing [12–14]. In this work, we use the Percus–Yevick closure that belongs to a group of closures neglecting blocking term in the functions cij (r). The Percus–Yevick closure reads cij (r) = {exp[ − Uij (r) − 1}{1 + hij (r) − cij (r)} ; cim (r) = {exp[ − Uim (r) − 1}{1 + him (r) − cim (r)} ;
(6)
where the corresponding interactions have been given by Eq. (1). The 2D ROZ equations have been solved numerically by direct iterations. However, the MFP between colloids is of much interest, besides the pair correlation functions. It is de ned as usual [1–3], Wcc (R) = −ln[gcc (R)] = −ln[1 + hcc (R)] :
(7)
Let us now proceed with the description of the results obtained for the model in which the matrix species (quenched colloids) are of the diameter m = 7s . This value is relevant to the diameter of hard spheres used for the modelling of silica xerogel by Kaminsky and Monson and in the subsequent study of Vega et al. [17]. We have assumed that annealed colloids, free to equilibrate in the matrix, are ve times larger than solvent species, c = 5s . First, we choose the colloid density xed, ∗c = c c2 = 2:5 × 10−6 . The matrix density is taken to be low, m m2 = 0:098, i.e. the adsorbing medium microporosity is high, p =0:923. We consider the solvent density, ∗s = s = 0:3, and ∗s = 0:55; and investigate the pair distribution functions (pdf’s). The results of our calculations are presented in Fig. 1. The solvent–solvent, gss (R), and solvent–colloid, gsc (R), pdf ’s are shown in Fig. 1a. We observe that with increasing solvent density the function gss (R) behaves
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Fig. 1. The uid– uid, gss (R) and gsc (R); pdf ’s (part a), and uid-matrix pdf ’s, gsm (R) and gcm (R) (part b), are shown by solid and dotted lines, for adsorbed dispersion at ∗s = 0:55 and ∗s = 0:3; respectively. In part c the MFP, − Wcc (r), for dispersions at ∗s = 0:55; ∗s = 0:45; and ∗s = 0:3; is given by solid, dashed and dotted lines, respectively. All the results correspond to the PY closure for the ROZ equations. The parameters of the model are the following: ∗c = 2:5 × 10−6 ; ∗m = 0:098; m = 7; c = 5.
as in common, uncon ned uids – the oscillations re ecting packing e ects develop with increasing ∗s . It becomes more probable to nd solvent species on the surface of a colloid particle with increasing ∗s (Fig. 1a). The oscillations of the function gsc (R) re ect the formation of layered solvent structure on the surface of a colloid. Oscillations of both gss (R) and gsc (R) diminish in magnitude with increasing interparticle separation. These trends, in general, show that the presence of a low number of obstacles (matrix particles) does not contribute much to the structure of the dispersions in question. From the behavior of the solvent–matrix, gsm (R), and colloid–matrix, gcm (R), pdf ’s, see Fig. 1b, it follows that there exists quite high probability to nd either solvent or colloid species in contact with matrix particle (this probability is higher at a higher solvent
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density). However, these correlations decay as in usual mixtures, the cooperative e ect of a set of quenched matrix species on the pdf ’s cannot be elucidated from these results. Moreover, the behavior of the colloid–colloid MFP with increasing solvent density does not exhibit “abnormal” trends, Fig. 1c. The depletion-type attraction for colloids close to each other, becomes stronger with increasing ∗s . A set of repulsive barriers also develops. The oscillations of the MFP shift to smaller distances with increasing ∗s , similar to commonly studied colloidal dispersions [1–3]. To summarize, the e ect of highly diluted matrix species on both the pdf ’s and the MFP, for the model in question and for the set of parameters considered, is negligibly small if any. Due to the absence of reliable tools to evaluate the chemical potentials of species in colloidal dispersions adsorbed in di erent microporous media, we are not able to perform consistent comparisons of the structure in di erent adsorbents. Nevertheless, some useful conclusions about the structural properties in terms of the pdf ’s, and of the MFP from the ROZ equations, can be made. In Fig. 2 we present the results of our calculations for the dispersion at ∗c =c c2 =2:5×10−6 , ∗s =0:3; in disordered matrices at m m2 = 0:098(p = 0:923); m m2 = 0:294(p = 0:769); and at m m2 = 0:441(p = 0:654). First, it is worth mentioning that the shape of the solvent–solvent, gss (R), and solvent– colloid, gsc (R), pdf’s presented in Fig. 2a, is very di erent, in comparison to the one given in Fig. 1a. Most important is that the dispersion is strongly con ned in microporous matrices with p = 0:769 and with p = 0:654; we do not observe monotonously decaying oscillations with increasing interparticle separation. Rather, the presence of strong con nement results in a high probability to nd particles close to each other. The extended minima, seen in gss (R) and gsc (R) at large separations, are due to the presence of matrix species with the diameter m = 7. The unusual shape, in comparison to the one shown in Fig. 1b, is observed for the solvent–matrix, gsm (R), and colloid– matrix, gcm (R), pdf’s, Fig. 2b. The colloid–colloid correlations in a strongly con ned dispersion (Fig. 2c) also di er from those shown in Fig. 1c. The modulating e ect of the solvent species on the MFP at small intercolloid separation is well pronounced. On the other hand, there appears an additional repulsive barrier on the MFP at large distances, due to the matrix species present. Modulation of the MFP by solvent particles also has been observed at large distances, these oscillations are either due to solvents on the surface of a matrix particle or on the surface of a colloid particle. However, from a comparative analysis of the gsm (R) and of the gsc (R); it is dicult to attribute solvents either to a layer covering matrix species or to a layer covering colloids. A speci c shape of the mean force potential acting between colloidal species in adsorbed dispersion in a microporous media is, however, one of our principal ndings. Several important issues remain to be investigated along the initiated line of studies. In particular, reliable tools to calculate thermodynamic properties of adsorbed dispersions must be developed. This would permit us to get an insight into possible mechanism of demixing of components in a highly asymmetric adsorbed mixtures; integral equations, unfortunately, do not represent adequate tools in this respect. Nevertheless, it seems to be of much interest to extend the model in question for systems with long-range, decaying repulsive potentials of Yukawa type, and investigate the mean
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Fig. 2. The same as in Fig. 1, but for the dispersion at ∗s = 0:3; ∗c = 2:5 × 10−6 , adsorbed in matrices at density ∗m = 0:098 (dotted lines), ∗m = 0:294 (dashed lines), and ∗m = 0:441 (solid lines).
force potential. Replica OZ equations and a wider repertoire of closures would be helpful for this purpose. Our preliminary calculations by using the hypernetted chain closure however, show, that trends observed in the present study are preserved. On the other hand, development of computer simulation algorithms for the systems in question remains a real challenge. This work has been supported in parts by the DGAPA of the National University of Mexico (UNAM) under Research Project IN 111597 and by the National Council for Science and Technology (CONACyT) of Mexico under project 25301-E. References [1] X.L. Chu, A.D. Nikolov, D.T. Wasan, Chem. Eng. Comm. 148–150 (1996) 123. [2] X.L. Chu, A.D. Nikolov, D.T. Wasan, J. Chem. Phys. 103 (1995) 6653.
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